Characterization of Highly Robust Solutions in Multi-Objective Programming in Banach Spaces

Abstract: This paper delves into the challenging issues in uncertain multi-objective optimization, where uncertainty permeates nonsmooth nonconvex objective and constraint functions. In this context, we investigate highly robust (weakly efficient) solutions, a solution concept defined by efficiency across all scenarios. Our exploration reveals important relationships between highly robust solutions and other robustness notions, including set-based and worst-case notions, as well as connections with proper and isolated efficiency. Leveraging modern techniques from variational analysis, we establish necessary and sufficient optimality conditions for these solutions. Moreover, we explore the robustness of multi-objective optimization problems in the face of various uncertain sets, such as ball, ellipsoidal, and polyhedral sets.